Cocycles for Kottwitz cohomology

TL;DR

This paper develops a framework for algebraic Kottwitz cocycles, providing finer control over cohomology sets related to Galois gerbes, which is essential for applications in Shimura varieties.

Contribution

It introduces a new framework for working with algebraic Kottwitz cocycles, enhancing the understanding of Galois gerbes in cohomology.

Findings

Framework enables explicit handling of algebraic cocycles

Facilitates applications to Shimura varieties

Provides finer control over cohomology sets

Abstract

In his work on defining the pointed set B(G) for all local and global fields, Kottwitz introduced certain Galois gerbes and considered their 'algebraic' cohomology with values in algebraic groups. However, the gerbes so constructed are only canonical up to conjugation by their bands. This is enough in order to ensure that the cohomology set B(G) is canonical, but not enough data to pin down the set of algebraic cocycles used to compute the set B(G). For arithmetic applications it is often desirable to have the finer control provided by these cocycles readily at hand. To this end we propose in this paper a framework with which to work with such spaces of algebraic Kottwitz cocycles. This will play a crucial role in our forthcoming work on the formalism of Shimura varieties.